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28 III METRIC SPACES

^&i general topology," as developed for instance in Kelley [15] and Bourbaki
[5]; the way to this generalization is made apparent in the remarks of Section
(3.11) when it is realized that in most questions, the distance defining a metric
space only plays an auxiliary role, and can be replaced by " equivalent" ones
without disturbing in an appreciable way the phenomena under study. In
Chapter XII, we shall develop the notions of general topology which will be
needed in further chapters.

I, DISTANCES AND METRIC SPACES

Let E be a set. A distance on E is a mapping d of E x E into the set R of
real numbers, having the following properties:

(I) d(x, y) ^ 0 for any pair of elements x₉ y of E.

(II) The relation d(x, y) = 0 is equivalent to x = y.

(III) d(y, x) = d(x, y) for any pair of elements of E.

(IV) d(x,z)^d(x,y) + d(y,z) for any three elements x,y, z of E
(" triangle inequality ").

From (IV) it follows by induction that

d(x_ly x_n) <dfa, x₂) + d(x₂, *₃) + • • • + d(x_n._l9 x_n)
for any n > 2.

(3.1.1) If d is a distance on E, then

\d(x,z)-d(y,z)\^d(x,y)
for any three elements x, y, z ofE.

For it follows from (III) and (IV) that

d(x, z) < d(y, z) + d(x₉ y)
and

d(y, z) < d(y, x) + d(x, z) = d(x, y) + d(x, z}
hence

-<*(*, JO < d(x, z) - d(y, z) ^ d(x, y).



	28 III METRIC SPACES ^&i general topology," as developed for instance in Kelley [15] and Bourbaki [5]; the way to this generalization is made apparent in the remarks of Section (3.11) when it is realized that in most questions, the distance defining a metric space only plays an auxiliary role, and can be replaced by " equivalent" ones without disturbing in an appreciable way the phenomena under study. In Chapter XII, we shall develop the notions of general topology which will be needed in further chapters.

	I, DISTANCES AND METRIC SPACES Let E be a set. A distance on E is a mapping d of E x E into the set R of real numbers, having the following properties: (I) d(x, y) ^ 0 for any pair of elements x₉ y of E. (II) The relation d(x, y) = 0 is equivalent to x = y. (III) d(y, x) = d(x, y) for any pair of elements of E. (IV) d(x,z)^d(x,y) + d(y,z) for any three elements x,y, z of E (" triangle inequality "). From (IV) it follows by induction that d(x_ly x_n) <dfa, x₂) + d(x₂, ₃) + • • • + d(x_n._l9 x_n)* for any n > 2. (3.1.1) If d is a distance on E, then \d(x,z)-d(y,z)\^d(x,y) for any three elements x, y, z ofE. For it follows from (III) and (IV) that d(x, z) < d(y, z) + d(x₉ y) and d(y, z) < d(y, x) + d(x, z) = d(x, y) + d(x, z} hence -<(, JO < d(x, z) - d(y, z) ^ d(x, y).